## MAT 530 Topology, Geometry I, Fall 2009.

### The final exam is on Wednesday, Dec 16, 2:15-4:45 in Physics P124.

• Instructor: Olga Plamenevskaya, office 3-107 Math Tower, e-mail: olga@math.sunysb.edu
• Office hours: MF 10:40am-11:40pm in P-143 or by appointment.
• Grader: Alexandra Popa, office 3-105 Math, email: alexandra@math.sunysb.edu,
• Class meetings: Monday and Friday, 12:50-2:10pm, Physics P124.
• References :
James R. Munkres, Topology, 2nd edition.
Munkres's book is available in the campus bookstore (but can certainly be found for less money elsewhere). It is the required text. Parts of the homework will be assigned from it, and there will be required readings.

Other useful books:

Allen Hatcher, Algebraic Topology. This book is available for free from Hatcher's webpage; Chapter 1 that covers the fundamental group is here.

William S. Massey, Algebraic Topology: An Introduction, GTM 56. (The first few chapters are also contained in another book by Massey, GTM 127). Massey has a nice treatment of classification of surfaces.

Important: Please write up your solutions neatly, be sure to put your name on them and staple all pages. Illegible homework will not be graded. Late homework will not be accepted. You are welcome to collaborate with others and even to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.

Homework 1: Munkres §13: 1, 5 (basis part only), 8 and five more questions (pdf), due Sept 11 .
Homework 2: Munkres §16: 8 ; §17: 6, 7, 19, and three more questions (pdf), due Sept 18. Required Reading: please read §18.
Homework 4: pdf, due Oct 2.
Homework 5: pdf, due Oct 9.
Homework 6: pdf, due Oct 16. 10/13: mistake in Problem 1 corrected.
Homework 7: pdf, due Oct 23.
Homework 8: pdf, due Nov 6.
Homework 9: pdf, due Nov 13.
Homework 10: pdf, due Nov 20.
Homework 11: pdf, due Dec 4. Please start early, it's a big one!
Homework 12: pdf, due Dec 11. The very last one!
Some solutions (or at least some hints) pdf,

• Syllabus: we will follow the basic outline from the graduate core course requirements (see below), not necessarily in the same order, with additional topics as time permits.
1. Basic point set topology
• Metric Spaces
• Topological spaces and continuous maps
• Comparison of topologies
• Separation axioms and limits
• Countability axioms, the Urysohn metrization theorem
• Compactness and paracompactness, the Tychonoff theorem
• Connectedness
• Product spaces
• Function spaces and their topologies, Ascoli's theorem
2. Introduction to algebraic topology
• Fundamental group
• Fundamental group of Sn; examples of fundamental groups of surfaces
• Seifert-van Kampen theorem
• Classification of covering spaces, universal covering spaces; examples
• Homotopy; essential and inessential maps

Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.